## Taiwanese Journal of Mathematics

### SOME FAMILIES OF INFINITE SERIES SUMMABLE BY MEANS OF FRACTIONAL CALCULUS

#### Abstract

In a Five-volume work published recently, K. Nishimoto [1] has presented a systematic account of the theory and applications of fractional calculus in a number of areas (such as ordinary and partial differential equations, special functions, and summation of series). In 2001, K. Nishimoto, D.-K. Chyan, S.-D. Lin and S.-T. Tu [11] derived the following interesting families of infinite series via fractional calculus, $$\displaystyle\sum_{k=2}^{\infty}\ \frac{(-c)^k}{k(k-1)}\frac{(kz-c)}{(z-c)^{k-1}}=c^2\ \biggr(\biggr|\displaystyle\frac{-c}{z-c}\biggr|\lt 1\biggr).$$ The object of the present paper is to extend the above families of infinite series to more general closed form relations. Various numerical results are also provided.

#### Article information

Source
Taiwanese J. Math., Volume 6, Number 4 (2002), 465-474.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500407471

Digital Object Identifier
doi:10.11650/twjm/1500407471

Mathematical Reviews number (MathSciNet)
MR1937472

Zentralblatt MATH identifier
1027.26006

#### Citation

Nishimoto, K.; Tu, Shih-Tong; Chen, I-Chun. SOME FAMILIES OF INFINITE SERIES SUMMABLE BY MEANS OF FRACTIONAL CALCULUS. Taiwanese J. Math. 6 (2002), no. 4, 465--474. doi:10.11650/twjm/1500407471. https://projecteuclid.org/euclid.twjm/1500407471